A Fast Earth-scattering Formalism for Light Dark Matter with Dark Photon Mediators

TL;DR

The paper tackles the challenge of modeling Earth-scattering effects for MeV-scale dark matter with a dark photon mediator, where scattering in the atmosphere and Earth's crust can imprint a daily modulation on the signal and potentially mimic backgrounds. It introduces Verne2, a fast semi-analytic formalism that accounts for attenuation and reflection of DM (up to two scatters) along straight-line trajectories, enabling efficient predictions of the velocity distribution at detectors. The method is validated against full Monte Carlo simulations (DaMaSCUS), achieving typical accuracies of $\mathcal{O}(10\%)$–$\mathcal{O}(30\%)$ with a speed-up of about $\sim 10^4$, making it practical for systematic DM parameter-space exploration and modulation studies. The approach, grounded in dark photon mediator phenomenology and including nuclear screening effects, supports rapid signal modeling for current and near-future DM-electron searches (e.g., DAMIC-M, SENSEI) and enhances the ability to leverage diurnal modulation as a robust background discriminator in the MeV mass range.

Abstract

While Dark Matter (DM) is typically assumed to interact only very weakly with the particles of the Standard Model, many direct detection experiments are currently exploring regions of parameter space where DM can have a large scattering cross section. In this scenario, DM may scatter in the atmosphere and Earth before reaching the detector, leading to a distortion of the DM flux and a daily modulation of the signal rate as the detector is shielded by more or less of the Earth at different times of day. This modulation is a distinctive signature of strongly-interacting DM and provides a powerful method of discriminating against time-independent backgrounds. However, the calculation of these Earth-scattering effects by Monte Carlo methods is computationally intensive, inhibiting a systematic exploration of the DM parameter space. Here, we present a semi-analytic formalism for calculating Earth-scattering effects, for models of MeV-mass DM which interacts via a dark photon mediator, and release the associated code Verne2. This formalism assumes that DM travels along straight-line trajectories until it scatters and is reflected back along its incoming path, along us to taking into account the affects of both attenuation and reflection in the Earth. We compare this formalism with the results of full Monte Carlo simulations for cross sections within reach of current and future DM-electron scattering searches. We find that Verne2 is accurate to better than 10-30%, making it suitable for performing signal modeling in the search for daily modulation, while reducing the computational cost by a factor of $\sim10^4$ compared to full Monte Carlo simulations.

Figures (9)

• Figure 1: Geometry of Earth-scattering Effects. The mean DM velocity is fixed by $\langle \mathbf{v}_\chi \rangle = -\mathbf{v}_E$, where $\mathbf{v}_E$ is the velocity of the Earth in the rest-frame of the Milky Way. The distortion to the DM velocity distribution due to Earth-scattering depends on the isodetection angle $\gamma$ between the mean DM velocity and the local zenith of the detector (an alternative definition for the isodetection angle is $\Theta = 180^\circ - \gamma$Emken:2017qmp). As the Earth rotates, the detector traces different values of $\gamma$, leading to a daily modulation of the DM signal rate.
• Figure 2: Total DM-nucleus scattering cross section for dark photon mediated interactions, relative to the unscreened case. We assume scattering off Oxygen nuclei. In the ultra-light mediator case (orange), the total cross section saturates to $(a\,q_\mathrm{ref})^{4}$ times the unscreened cross section at low velocity (we have factored this numerical value out for the ultra-light mediator, leading to saturation at 1). In the heavy mediator case (blue), the total cross section is zero at low velocity, saturating to the unscreened value at large velocities.
• Figure 3: Distribution of the deflection angle $\alpha$. Note that $\cos\alpha = 1$ corresponds to forwards scattering while $\cos\alpha$ corresponds to backwards scattering.
• Figure 4: Scheme for attenuation and reflection of DM particles. (a) DM particle path to the detector with 0-scatters and 1-scatter. The unscattered particle will traverse $L_1$ and reach the detector. Once it passes the detector, the particle can be reflected back again along the path $L_2$. (b) 2-scatter scheme of DM through the Earth. The dashed arrows represents the trajectory of the particle after backwards scatter at $x_1$ anywhere within the path $L_1$, followed by a backwards scatter at $x_2$ anywhere within the path $L_1$ before the detector.
• Figure 5: Velocity distribution correction due to Earth scattering $p(v,\,\theta)$, for a given DM velocity $v$ and incoming direction $\theta$. Particles arriving from below the detector (and crossing most of the Earth) have $\theta = 0^\circ$. For both the ultra-light mediator (left) and the heavy mediator (right), we fix the DM mass and cross section to be $m_\chi = 1\,\mathrm{MeV}$ and $\bar{\sigma}_p = 10^{-32}\,\mathrm{cm}^2$.